Tangents to a curve as limiting case of secants

The tangent to a curve at a point as a limiting case of a secant through this point

The construction is the same on the three curves (all in blue): 1. A point is given (in red) 2. The tangent to the curve is plotted (dashed, in red) 3. Through the given point, a secant to the curve is plotted, with variable slope t (see slider) 4. Look at the points of intersection of the secant with the curve 5. Using the slider, look at what happens to these points when t varies. Pay attention: (i) The point of tangency can be viewed as a "double point", but it can be also a"triple point" or even a point of higher multiplicity. (ii) For curves of degree at least 3, a tangent can intersect the curve at another point.